To determine the original coordinates of point [tex]\( K \)[/tex] given its translated image [tex]\( K^{\prime}(-8, 6) \)[/tex] and the translation rule [tex]\((x, y) \rightarrow (x - 3, y + 6)\)[/tex], follow these steps:
1. Understand the translation rule:
- The rule [tex]\((x, y) \rightarrow(x - 3, y + 6)\)[/tex] indicates that for any point [tex]\((x, y)\)[/tex], the x-coordinate is decreased by 3 and the y-coordinate is increased by 6.
2. Set up the relationship using the given translation rule:
- For the image [tex]\( K^{\prime}(x', y') \)[/tex] of a point [tex]\( K(x, y) \)[/tex], the coordinates transformation is:
[tex]\[
x' = x - 3 \quad \text{and} \quad y' = y + 6
\][/tex]
3. Substitute the given coordinates of [tex]\( K^{\prime} \)[/tex] into the transformation equations:
- Given [tex]\( K^{\prime}(-8, 6) \)[/tex]:
[tex]\[
x' = -8 \quad \text{and} \quad y' = 6
\][/tex]
4. Solve for the original coordinates [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- Using [tex]\( x' = -8 \)[/tex]:
[tex]\[
-8 = x - 3
\][/tex]
Add 3 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = -8 + 3 = -5
\][/tex]
- Using [tex]\( y' = 6 \)[/tex]:
[tex]\[
6 = y + 6
\][/tex]
Subtract 6 from both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 6 - 6 = 0
\][/tex]
5. Combine the results:
- The original coordinates of point [tex]\( K \)[/tex] are:
[tex]\[
K = (-5, 0)
\][/tex]
Therefore, the coordinates of [tex]\( K \)[/tex] are [tex]\((-5, 0)\)[/tex].
